Respiratory rate can provide auxiliary information on the physiological changes within the human body, such as physical and emotional stress. In a clinical setup, the abnormal respiratory rate can be indicative of the deterioration of the patient's condition. Most of the existing algorithms for the estimation of respiratory rate using photoplethysmography (PPG) are sensitive to external noise and may require the selection of certain algorithm-specific parameters, through the trial-and-error method. This paper proposes a new algorithm to estimate the respiratory rate using a photoplethysmography sensor signal for health monitoring. The algorithm is resistant to signal loss and can handle low-quality signals from the sensor. It combines selective windowing, preprocessing and signal conditioning, modified Welch filtering and postprocessing to achieve high accuracy and robustness to noise. The Mean Absolute Error and the Root Mean Square Error of the proposed algorithm, with the optimal signal window size, are determined to be 2.05 breaths count per minute and 2.47 breaths count per minute, respectively, when tested on a publicly available dataset. These results present a significant improvement in accuracy over previously reported methods. The proposed algorithm achieved comparable results to the existing algorithms in the literature on the BIDMC dataset (containing data of 53 subjects, each recorded for 8 min) for other signal window sizes. The results endorse that integration of the proposed algorithm to a commercially available pulse oximetry device would expand its functionality from the measurement of oxygen saturation level and heart rate to the continuous measurement of the respiratory rate with good efficiency at home and in a clinical setting. The online version contains supplementary material available at 10.1007/s40846-022-00700-z.